Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 2: Numerical...

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2002; 55:253–291 (DOI: 10.1002/nme.493)

Classical and advanced multilayered plate elements basedupon PVD and RMVT. Part 2: Numerical implementations

Erasmo Carrera∗;† and Luciano Demasi

DIASP; Politecnico di Torino; Torino; Italy

SUMMARY

This paper presents numerical evaluations related to the multilayered plate elements which were proposedin the companion paper (Part 1). Two-dimensional modellings with linear and higher-order (up to fourthorder) expansion in the z-plate=layer thickness direction have been implemented for both displacementsand transverse stresses. Layer-wise as well as equivalent single-layer modellings are considered on bothframeworks of the principle of virtual displacements and Reissner mixed variational theorem. Sucha variety has led to the implementation of 22 plate theories. As far as �nite element approximationis concerned, three quadrilaters have been considered (four-, eight- and nine-noded plate elements).As a result, 22×3 di�erent �nite plate elements have been compared in the present analysis. Theautomatic procedure described in Part 1, which made extensive use of indicial notations, has hereinbeen referred to in the considered computer implementations. An assessment has been made as faras convergence rates, numerical integrations and comparison to correspondent closed-form solutionsare concerned. Extensive comparison to early and recently available results has been made for sampleproblems related to laminated and sandwich structures. Classical formulations, full mixed, hybrid, aswell as three-dimensional solutions have been considered in such a comparison. Numerical substantiationof the importance of the ful�lment of zig-zag e�ects and interlaminar equilibria is given. The superiorityof RMVT formulated �nite elements over those related to PVD has been concluded.Two test cases are proposed as ‘desk-beds’ to establish the accuracy of the several theories. Results

related to all the developed theories are presented for the �rst test case. The second test case, which isrelated to sandwich plates, restricts the comparison to the most signi�cant implemented �nite elements.It is proposed to refer to these test cases to establish the accuracy of existing or new higher-order,re�ned or improved �nite elements for multilayered plate analyses. Copyright ? 2002 John Wiley &Sons, Ltd.

KEY WORDS: �nite element; plates; multilayers; classical and mixed formulation; composite materials

1. INTRODUCTION AND CONTENTS OF THE PAPER

This paper quotes numerical evaluations of the multilayered �nite plate elements that havebeen proposed in the companion paper (Part 1) [1]. Reissner’s mixed variational theorem

∗Correspondence to: Erasmo Carrera, Department of Aeronautics and Aerospace Engineering, Politinecnico diTorino, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy.

†E-mail: [email protected] 23 October 2000

Copyright ? 2002 John Wiley & Sons, Ltd. Revised 4 April 2001

254 E. CARRERA AND L. DEMASI

(RMVT) and the principle of virtual displacements (PVD) have been used to developadvanced and classical plate elements, respectively. Both layer-wise (LW) and equivalentsingle-layer (ESL) variable descriptions were employed (it is intended that the number ofthe unknown variables remains independent of the number of constitutive layers in the lattercase).The number of nodes Nn of the elements, as well as the order of the expansion N for both

displacement and transverse stress variables in the plate thickness direction z, were taken asfree parameters in the developments documented in Part 1. The numerical implementations,whose results are documented in this paper, have been restricted to the following cases:

1. Three quadrilater cases have been implemented, as far as the number of nodes isconcerned: Nn=4,8 and 9 (that is, four-, eight- and nine-noded plate elements havebeen considered)

2. N64, that is, theories which permit linear up to fourth-order expansion for displacementand=or transverse stress variables in the layer or whole plates have been addressed.

The latter choice, in conjunction with the possibilities of choosing between PVD or RMVTimplementation, as well as layer-wise or equivalent single-layer description, lead to 22 possibletwo-dimensional theories. Referring to point (1), it follows that 22×3, multilayered �niteelements have been implemented and compared in this paper. Some of these are classical�nite elements (those based on PVD and linear �eld in layer=plate thickness). Nevertheless,about 20×3 of the multilayered plate elements discussed in this paper have neither beenproposed nor evaluated by other scientists. These elements, herein called advanced, which areformulated on the basis of Reissner’s mixed variational theorem are of particular interest.The 22×3 �nite elements have been implemented in a �nite element code which is available

at the DIASP. The possibilities of implementing all these elements in a single step is closelyrelated to the following crucial facts, which were described in Part 1:

• the extensive use of indicial notations;• the reduction of each type of �nite element matrix to only �ve arrays (of dimension 3×3),which have already been called ‘fundamental nuclei’ (one for PVD formulations, seeEquation (58) of Part 1, and four for RMVT ones, see Equations (69)–(72) of Part 1).

As a result, it was possible to implement the 22×3 multilayered plate elements, all together,by putting only 5×9 lines in appropriate loops of the written FORTRAN code. For the sakeof clarity, these FORTRAN lines have been explicitly written in Appendix A.The authors have found this technique very convenient and easy to check, and encourage

other scientists to refer to it in the case where they are going to implement �nite elementsthat are somewhat similar to the authors’ one.In spite of the large number of implemented elements, not all the possibilities described

in Part 1 have herein been addressed. For instance, full mixed implementation has not beendiscussed. Mixed implementations have therefore been restricted to the case in which stressvariables are eliminated at the element level; interlaminar loadings have not been addressedand so on. However, further implementations are in progress and the related results could bethe subject of future studies.The contents of this paper have been organized as follows. Section 2 quotes details on the

implemented �nite elements. The used shape functions, as well as displacement and transversestress �elds in the thickness directions, are given and explained with the help of �gures.

Copyright ? 2002 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2002; 55:253–291

PLATE ELEMENTS BASED UPON PVD AND RMVT. PART 2 255

Acronyms have been introduced in this section to denote the implemented �nite elements.Section 3 describes the conducted numerical investigations. The data of the treated problemsare �rstly quoted. An assessment and comparison to other available results are then given insubsequent subsections. Section 4 discusses two test cases which are proposed as desk-beds toassess multilayered �nite plate elements. Section 5 traces the �nal remarks. The FORTRAN listof the �ve used fundamental nuclei are explicitly given and brie�y explained in Appendix A.As far as notation and symbols are concerned, full reference should be made to Part 1.

2. IMPLEMENTED FINITE ELEMENTS

Apart from the number of nodes, the implemented multilayered �nite elements are character-ized by

• order N of the expansion in the z-thickness direction;• used variational statement (PVD or RMVT);• variables description (LWM or ESLM).

In order to denote in a concise manner the implemented �nite elements, acronyms could beconveniently used. These acronyms have herein been built as illustrated in Figure 1:

• The �rst �eld of the acronym is related to the used variables description, L states forlayer-wise and E states for equivalent single layer.

• The second �eld signi�es the variational statements on which the correspondent �niteelement relay, D and M mean Principle of Virtual Displacements and Reissner’s MixedVariational theorem, respectively.

• The third and fourth �elds (which are optional in the sense that they do compare onlyin particular cases related to ESL description) can assume the letter Z or=and C; that iszig-zag function is included in the displacement �eld and=or interlaminar continuity isguaranteed for transverse stresses.

• The last �eld is a number which denotes the used values for N .Extensive reference to such acronyms will be made in the following.

2.1. Used �nite element approximations

Following standard �nite element method, the unknown variables in the generic layer pointsof the reference surface � are expressed in terms of their values with correspondence to thenodes. According to an isoparametric description displacements are written as follows:

uk� =Niqk�i (i=1; 2; : : : ; Nn) (1)

where Ni denote the shape functions and

qk�i=[qkux�i q

kuy�i q

kuz�i]

T (2)

are the nodal values of displacements.The same approximations have been introduced for stresses,

�kn�=Nigk�i (i=1; 2; : : : ; Nn) (3)



L

E

D

M

1

2

3

4

Z C

TYPE OF FORMULATION

Classical based on PVD

Mixed based on RMVT

D

M

Linear

Parabolic

Cubic

Fourth-order

z-EXPANSION FOR ORDER OF USED

OPTIONAL FIELDS USED FOR ESLM MODELS

Zig-Zag Effects is Accounted forZ

C Interlaminar Equilibria is fulfilled

ACRONYM

Equivalent-Single-LayerE

Layer-WiseLTYPE OF THEORY

Layer-Wise Theory based on Classical Displacement formulation with cubic displacement fields in the layer

EMZC2

LD3

EXAMPLES

Mixed Equivalent-Single-Layer with parabolic displacement fields (and cubic stress fields) accounting for Zig-zag Effectand fulfilling interlaminar transverse stresses Continuity

Figure 1. Acronyms used to denote implemented �nite elements.

where

gk�i=[gkxz�i g

kyz�i g

kzz�i]

T (4)

Three plate elements have been implemented with four (Q4), eight (Q8), nine (Q9) nodes,respectively. The explicit form of the shape function can be read in Reference [2]. Accordingto isoparametric description, the problem co-ordinates are given in terms of their nodal valuesXi and Yi, according to

x=Nn∑1Ni(�; �)Xi; y=

Nn∑1Ni(�; �)Yi

2.2. Multilayered plate elements based on PVD

Particular cases of displacement and stress �elds which were introduced in Sections 5 and 6of Part I are explicitly described here.First, multilayered �nite elements based on PVD are discussed. In such a framework, only

displacements are assumed.First- and higher-order ESLM cases: ED1, ED2, ED3, ED4: The order of the Taylor-type

expansion varies from 1 up to 4. ESLM description requires a unique displacement �elds forthe whole multilayered plate,

u= u0 + zrur ; r=1; 2; : : : ; N (5)



z

x,y

z

x,y

Figure 2. ESLM assumption. Linear and cubic cases.

Subscript 0 denotes displacement values with correspondence to the plate reference surface �.Linear and higher-order distributions in the z-direction are introduced by the r-polynomials.In a uni�ed notation, the above expansion is rewritten in the following manners

u=Ftut + Fbub + Frur =F�u�; �= t; b; r; r=1; 2; : : : ; N − 1 (6)

where

Fb=1; Ft = zN ; Fr = zr; r=1; 2; : : : ; N − 1 (7)

b and t subscripts in the LW cases will signify, see below, values of the displacement and=orstress variables with correspondence to layer bottom and top surfaces, respectively.By varying N , from 2 to 4, four sub-cases are obtained. According to the acronyms of

Figure 1, these four cases correspond to ED1, ED2, ED3, ED4 (E signi�es equivalent singlelayer, D states that the related �nite elements are formulated with only displacement variablesand based on PVD, the �nal numbers denote the order of the used expansion). Linear andcubic cases are depicted in Figure 2.First- and higher-order ESL cases with zig-zag function: EDZ1, EDZ2, EDZ3: zig-zag

e�ect can be introduced in the �nite element models with only displacement variables alongwith ESLM description, by referring to Murakami’s zig-zag function. According to Murakami[3], a zig-zag function is added to the displacement �eld related to ED cases, as it follows

u= u0 + (−1)k�kuZ + zrur ; r=1; 2; : : : ; N (8)

Subscript Z refers to the introduced zig-zag term. Note that the unknown variables u0; uZ ; ur arek-independent. That is, ESLM description has been preserved. �k =2zk=hk is a non-dimensionallayer co-ordinate (zk is the physical co-ordinate of the k-layer whose thickness is hk). Theexponent k changes the sign of the zig-zag term in each layer.



z

x,y

ED1

Zig-Zag

z

x,y

Zig-Zag

ED3

Figure 3. ESLM assumption with zig-zag function. Linear and cubic cases.

By employing a uni�ed notations Equation (8) becomes

u=Ftut + Fbub + Frur =F�u�; �= t; b; r; r=1; 2; : : : ; N (9)

Subscript t has been chosen to denote the zig-zag term (ut = uZ , Ft = (−1)k�k).The following three �nite element implementations have been considered: EDZ1, EDZ2,

EDZ3. The letter Z , before the order of the expansion, signi�es that zig-zag e�ect is accountedfor in these �nite elements. The linear and cubic cases are depicted in Figure 3. The geomet-rical meanings of the zig-zag function become clear by the quoted picture: a zig-zag functionof constant amplitude, which changes its sign with correspondence to each interface, is addedto classical displacement �eld of ED-type. As a result, linear and higher-order displacement�elds are obtained which reproduce derivative discontinuity at the interfaces.First- and higher-order LW elements: LD1, LD2, LD3, LD4: Linear and higher-order

displacement �elds can be employed in the framework of layer-wise description. That is, thedisplacement �eld at Equation (6) is taken in each layer, in formulae

uk =Ftukt + Fbukb + Fru

kr =F�u

k� ; �= t; b; r; r=2; 3; : : : ; N; k=1; 2; : : : ; Nl (10)

It is now intended that the subscripts t and b denote values related to the layer top andbottom surfaces, respectively. These two terms consist of the linear part of the expansion.The thickness functions F�(�k) have now been de�ned at the k-layer level,

Ft =P0 + P12

; Fb=P0 − P12

; Fr =Pr − Pr−2; r=2; 3; : : : ; N (11)

in which Pj=Pj(�k) is the Legendre polynomial of j-order de�ned in the �k-domain: −16�k61. The �rst �ve used Legendre polynomials are

P0 = 1; P1 = �k ; P2 = (3�2k − 1)=2; P3 =5�3k2

− 3�k2; P4 =

35�4k8

− 15�2k4+38



z

x,y

z

x,y

Figure 4. LWM assumption. Linear and cubic cases.

The chosen functions have the following properties:

�k =

{1; Ft =1; Fb=0; Fr =0

−1; Ft =0; Fb=1; Fr =0(12)

The continuity of the displacement at each interface is easily linked:

ukt = u(k+1)b ; k=1; Nl − 1 (13)

The four �nite elements LD1, LD2, LD3, LD4 have been implemented. L signi�es layer-wisedescription, D states that the related �nite elements are formulated with only displacementvariables and based on PVD, the �nal numbers denote the order of the used expansion. Linearand cubic cases have been depicted in Figure 4.

2.3. Plate elements based upon RMVT

The �nite elements described above are not able to ful�l completely and a priori theinterlaminar equilibria for the transverse stresses. In order to meet such a requirement, stressassumptions should be made. According to Part 1, the description interlaminar continuoustransverse shear and normal stresses require layer-wise modellings. Therefore, ESL descrip-tion could be only given for displacement variables. Implemented �nite elements which arebased on di�erent displacement and stress assumptions and on the application of RMVT aredescribed in this subsection.First- and higher-order ESLM cases: EMC1, EMC2, EMC3, EMC4: Equivalent single-

layer �nite elements based on RMVT (as discussed in Part 1) can be developed by using thedisplacement �elds, already described in Equation (6). Such a �eld is herein, for convenience,rewritten as

u=Ftut + Fbub + Frur =F�u�; �= t; b; r; r=1; 2; : : : ; N − 1 (14)



Transverse shear and normal stresses are assumed according to the same assumption andnotation that have been used for displacements. In formulae,

�knM =Ft�knt + Fb�knb + Fr�knr =F��kn�; �= t; b; r; r=2; 3; : : : ; N; k=1; 2; : : : ; Nl (15)

The four cases EMC1, EMC2, EMC3, EMC4 have been implemented. E signi�es equivalentsingle layer, M states that the related �nite elements are formulated with mixed variables(stresses and displacements) and based on RMVT, C signi�es that transverse stresses area priori continuous at the interfaces; the �nal numbers denote the order of the used expansion.Linear and cubic cases have been depicted in Figure 2.First- and higher-order ESLM cases with zig-zag function: EMZC1, EMZC2, EMZC3:

The physics of multilayered structures demonstrated that interlaminar continuity and zig-zage�ects are strongly connected to each other, as it is in solids for equilibrium and compatibility.Interlaminar continuous transverse stresses not accompanied by zig-zag e�ects description, asdone for EMC-type �nite elements, is physically inconsistent. To remove such inconsistency,a zig-zag function is used in the RMVT and ESLM framework. As a result, the displacement�eld at Equation (8) is employed in conjunction with the transverse stress assumption ofEquation (15). Three multilayered �nite elements herein denoted by EMZC1, EMZC2, EMZC3have been implemented. With respect to that EMC case, a Z has been introduced in theacronyms to denote that zig-zag e�ects have been accounted for.Linear and cubic cases have been depicted in Figure 2.First- and higher-order LW cases: LM1, LM2, LM3, LM4: Full layer description based

upon RMVT is simply obtained by using layer-wise description for both displacement andtransverse stress variables. In formulae

uk = Ftukt + Fbukb + Fru

kr =F�u

k� ; �= t; b; r

r=2; 3; : : : ; N�knM = Ft�knt + Fb�knb + Fr�knr =F��kn�; k=1; 2; : : : ; Nl

(16)

The four �nite elements LM1, LM2, LM3, LM4 were implemented. Figure 4 depicts therelated displacement and stress �elds related to linear and cubic order cases.

3. RESULTS AND DISCUSSION

A large numerical investigation was conducted in order to assess the implemented �niteelements. A number of convergence studies and comparison to analytical solutions, to three-dimensional exact analyses and to other available �nite element results related to static re-sponse of bent orthotropic, multilayered very thick, thick, moderately thick and thin plateswere made. Di�erent loadings as well as boundary conditions were treated. Numerical perfor-mances have been established and transverse locking mechanisms were contrasted by extensiveuse of reduced and selective integration techniques. This was done according to the descriptiongiven in Appendix C of Part 1.

3.1. Data of the treated problems

Data of the considered problems are described herein. For convenience, laminated data andboundary conditions are denoted by acronyms.



Mechanical properties of the lamina and lay-out of the laminates: The mechanicalproperties of the used lamina are:

E1E2= 25;

G12E2=G13E2= 0:5;

G23E2= 0:2; �12 = �13 = �23 = 0:25

Cross-ply symmetrical and unsymmetrical laminated plates were investigated according to thefollowing lay-outs and thicknesses:

• LAM1: 0◦;• LAM2: 0◦=90◦=0◦, hi= 1

3h, i=1; : : : ; 3;• LAM3: 0◦=90◦=90◦=0◦, hi= 1

4h, i=1; : : : ; 4;• LAM4: 0◦=90◦=0◦=90◦, hi= 1

4h, i=1; : : : ; 4.

Further to the above four cross-ply laminates, a sandwich plate will be analysed in Section 4.The mechanical properties of the laminae which are used as skins are those by Pagano [4]:

EL =25; ET =1

GLT =0:5GTT =0:2�LT = �TT =0:25

The core material used for the sandwich plates is transversely isotropic with respect to thez-axis and is characterized by the following elastic properties:

Exx = Eyy=0:04; Ezz=0:5; Gxz=Gyz=0:04

Gxy =0:016; �xy= �zy= �zx=0:25

Loading cases: Four types of loadings have been treated (Figure 5). These are all appliedat the top surface of the investigated plates.

Figure 5. Considered loading cases. Concentrated load, constant and harmonic pressure distribution forcylindrical, bent and rectangular plates.



(1) Sinusoidal load:

Pz=pz sin(�xa

)(2) Bi-sinusoidal load:

Pz=pz sin(�xa

)sin

(�yb

)(3) Uniform pressure:

Pz=pz

(4) Point load applied at the plate centre:

Pz=P

Boundary conditions: As far as boundary conditions are concerned, reference will be madeto the two following cases:

• SS: simply supported;• CL: clamped.Adimensionalizations: Stresses and displacements are adimensionalized according to the

following formulas:

( ��xx; ��yy; ��xy) =1

pz(a=h)2(�xx; �yy; �xy)

( ��zx; ��zy) =1

pz(a=h)(�zx; �zy); ��zz=

1pz�zz

(�′xx; �′yy; �

′xy) =

1pz(�xx; �yy; �xy); (�′zx; �

′zy)=

1pz(�zx; �zy)

�Ux =uxE2pzh

; �Uz=uz100h3E2pza4

; U ′z =

uz100h3E2Pa2

3.2. Finite element assessment and comparison to correspondent closed-form solutions

The acronyms that have been introduced in the previous paragraphs are used in the tablesand �gures to denote di�erent theories as well as data description. Acronyms have also beenused to denote results by other authors. A list of these has been provided in Table I. If notdi�erently declared, it is intended that results are related to Q9 plate elements. Sinusoidaland bi-sinusoidal loadings are referred to cylindrical bending and square plate problems,respectively.Convergence rates, locking mechanism and comparison to correspondent analytical, closed-

form solution has been provided in Tables II–VI and Figures 6–22. The number of nodesNn, the number of elements in the meshes Ne and the number of expansions used in thez-direction N have been used as parameters in the made assessment. Locking problems have,



Table I. List of acronyms frequently used in tables and �gures to denote resultsfor multilayered plates taken by open literature.

3D Three-Dimensional solution (taken by di�erent sources)A&S Auricchio and Sacco [18]D&R Di and Ramm [19]D-1, D2 Linear and re�ned theory in Di Schiuva [20]H&L Jing and Liao [21]IK&T Idlbi, Karama and Touratier [22]L&S Liou and Sun [23]LH&X Liew, Han and Xiao [15]Mindlin Mindlin [24]Morya Morya [25]P&K Pandya and Kant [26]P&T Polit and Touratier [27]R–E Reddy [28]R-H Reddy [29]

Table II. �Uz (x= a=2; z=0). Convergence rate with respect to number ofelements. Data: IS integration, LAM2, sinusoidal load, SS.

a=h=4 a=h=6

Ne Q4 Q8 Q9 Q4 Q8 Q9

LD12 2.4900 2.8204 2.7798 1.3619 1.6026 1.58024 2.7104 2.8004 2.7838 1.5263 1.5952 1.58376 2.7514 2.7925 2.7839 1.5582 1.5900 1.58388 2.7658 2.7891 2.7839 1.5695 1.5877 1.583910 2.7725 2.7874 2.7839 1.5747 1.5864 1.5839LD1a 2.7830 1.5830

LM12 2.4886 2.8372 2.7866 1.3702 1.6218 1.59604 2.7154 2.8126 2.7915 1.5406 1.6130 1.59986 2.7580 2.8028 2.7918 1.5735 1.6071 1.59998 2.7729 2.7985 2.7918 1.5851 1.6044 1.599910 2.7799 2.7963 2.7918 1.5905 1.6029 1.5999LM1a 2.791 1.599

at this stage, been contrasted by the use of reduced=selective integration techniques. Thesehave been depicted in the correspondent analysis by adding the further letters IN, IS and IS2in brackets to the correspondent �nite elements. These integration schemes treat the di�erentsti�ness=compliance terms which were outlined in Appendix C of Part 1, in an alternate man-ner. The authors are not aware of the many other, more sophisticated, more elegant and moree�cient integration techniques such as those described by Bathe and Dvorkin [5] andalso used by the �rst author [6]. In fact, these advanced techniques permit one to avoidspurious modes that normally arise from reduced integration implementations. A discussionon these problems can be found in the articles by Briossilis [7–9], Zienkiewicz et al. [10]



Table III. �Uz (x= a=2; z=0). Convergence rate with respect to N . Data: IS integration,LAM2, sinusoidal load, SS, mesh 8×1.a=h=4 a=h=6

N Q4 Q8 Q9 LDa Q4 Q8 Q9 LDa

LDN case1 2.7658 2.7891 2.7839 2.783 1.5695 1.5877 1.5839 1.5832 2.8453 2.8700 2.8642 2.864 1.6156 1.6342 1.6302 1.6303 2.8693 2.8407 2.8878 2.887 1.6205 1.5983 1.6351 1.6354 2.8695 2.8410 2.8879 2.887 1.6206 1.5984 1.6351 1.6353D 2.887 1.635

LMN caseN Q4 Q8 Q9 LMa Q4 Q8 Q9 LMa

1 2.7729 2.7985 2.7918 2.791 1.5851 1.6044 1.5999 1.5992 2.8581 2.8277 2.8767 2.891 1.6182 1.5953 1.6328 1.6353 2.8693 2.8407 2.8878 2.887 1.6205 1.5983 1.6351 1.6354 2.8695 2.8410 2.8879 2.887 1.6206 1.6206 1.6351 1.6353D 2.887 1.635

Table IV. �Ux (x=0). Convergence to the elasticity solution by Pagano [30]. Data: a=h=4, LAM1,sinusoidal load, SS, mesh 8×1.

LM4 LM4 LM4 LD4a LD4 LD4z=h 3D LM4a (IN) (IS) (IS2) LD4 (IN) (IS) (IS2)

−0.5 0.6894 0.6855 0.6780 0.6880 0.6882 0.6855 0.6780 0.6880 0.6882−0.3 0.2260 0.2225 0.2197 0.2230 0.2228 0.2225 0.2197 0.2230 0.2228−0.1 0.0632 0.0520 0.0497 0.0503 0.0502 0.0520 0.0497 0.0503 0.05020.1 −0.0333 −0.0222 −0.0253 −0.0259 −0.0258 −0.0222 −0.0253 −0.0259 −0.02580.3 −0.2090 −0.2053 −0.2082 −0.2114 −0.2112 −0.2053 −0.2082 −0.2114 −0.21120.5 −0.7161 −0.7122 −0.7116 −0.7216 −0.7221 −0.7122 −0.7116 −0.7216 −0.7221

and Carrera [11]. On the other hand, this present paper is more oriented towards two-dimensional modellings of multilayered plates. In this respect, the previously mentioned ad-vanced techniques could be implemented in future works.The convergence rates of Q4, Q8 and Q9 elements have been documented in Table II and

Figures 6 and 7 with respect to the number of elements in a cylindrical bent plate. Two thickplates are considered. Both mixed (LM1) and classical (LD1) approaches are compared. Theanalytical closed-form solutions are taken from (Carrera’s) previous works [12–14]. The latterhave been denoted by adding superscript a to the correspondent �nite element acronyms. Anexcellent convergence rate has to be registered for both classical and mixed cases. It shouldbe noted that the comparison that has been made with the closed-form solutions consistsof the best test that could be made in order to assess the reliability of FE approximations.Finite element results could not in fact be better, in any case, than those of correspondingclosed-form solutions.



Table V. Convergence to the analytical solution. Data: a= b, LAM4, bi-sinusoidal load, SS, mesh 4×4.��xx ��yy ��xy ��zx ��zy ��zz �Uz

a=h a=2; a=2;± 12 a=2; a=2;± 1

4 0; 0;± 12 0; a=2; 0 a=2; 0; 0 a=2; a=2; 0 a=2; a=2; 0

2LM4a

LM4(IN)

0:1918−0:94470:2053

−0:9885

0:0265−0:78080:0260

−0:8147

−0:08770:0694

−0:09280:0733

0:1625

0:1727

0:1947

0:2058

0:4512

0:4512

5:2632

5:2642

10LM4a

LM4(IS)

0:04530:53600:0474

−0:5611

0:01140:49660:0121

−0:5239

−0:02910:2921

−0:03020:0304

0:2713

0:2638

0:2719

0:2645

0:4996

0:5015

0:7623

0:7629

100LM4a

LM4(IS2)

0:0358−0:48730:0377

−0:5111

0:0112−0:46270:0119

−0:4851

−0:25390:2541

−0:02620:0262

0:2803

0:3032

0:2803

0:3033

0:5000

0:5302

0:5092

0:5094

The following further comments can be made:

• Q9 elements show the best convergence rate.• As known, sti�ness reduces for Q4 and Q9 elements with Ne increasing, vice versa forthe Q8 case.

• Mixed (LM1) and classical elements (LD1) show the same numerical behaviour, thatis, RMVT does not introduce any complicating numerical e�ects to the correspondingPVD ones. This happens even with the quite di�erent mathematical structures of thedi�erential operators of RMVT matrices with respect to PVD ones.

Convergence to elasticity solutions and comparison to correspondent analytical ones have beenanalysed in Table III and Figure 8. The e�ect of an increase in N has been monitored. Amesh of eight elements has been used. Twenty four (2×4×3) multilayered �nite elements arecompared: four LD (LD4, LD3, LD2, LD1) and four LMN (LM4, LM3, LM2, LM1) casesfor each of Q4, Q8 and Q9 implementations. The following conclusions can be drawn:

• Finite element results closely match the correspondent analytical ones for each N valueand for both classical and mixed �nite elements.

• An increase of N the elasticity solutions is obtained.

The made analysis has demonstrated the suitability of the implemented elements. Furthermore,to obtain the desired accuracy and, at the same time, to cut down computational costs, acompromise has to be made between the number of used elements Ne as well as element typeNn, and the order of the used expansion for the variables in the plate thickness directions N .Further assessments which also take the implemented numerical integration schemes and

their e�ects into account on di�erent theories are documented in Tables IV–VII and Figures9–22. Comparison to three-dimensional solutions and=or to analytical ones have been providedin these analyses. Local values of transverse displacements, related to layer-wise descriptions,are reported in Table IV, for a thick plate. Finite elements and closed-form (analytical) solu-tions for transverse displacements are compared to three-dimensional ones in six positions of



Table VI. Convergence to the elasticity solution. Data: a= b, LAM3, bi-sinusoidal load, SS, mesh 4×4.��xx ��yy ��xy ��zx ��zy ��zz �Uz

a=h a=2; a=2;± 12 a=2; a=2;± 1

4 0; 0;± 12 0; a=2; 0 a=2; 0; 0 a=2; a=2; 0 a=2; a=2; 0

23D

LM4(IN)

1:388−0:9121:4405

−0:9477

0:835−0:7950:8478

−0:8116

−0:08630:0673

−0:09110:0710

0:153

0:1601

0:295

0:3105

—

0:4576

5:075

5:0800

43D

LM4(IN)

0:720−0:68400:7456

−0:7093

0:663−0:66600:6897

−0:6937

−0:04670:0458

−0:04930:0484

0:219

0:2294

0:292

0:3148

—

0:4964

1:937

1:9374

103D

LM4(IS)

0:559−0:5590:5909

−0:5915

0:401−0:4030:4225

−0:4244

−0:02750:0276

−0:02860:0287

0:301

0:3073

0:196

0:1607

—

0:5018

0:737

0:7376

203D

LM4(IS2)

0:543−0:54300:5732

−0:5741

0:308−0:30900:3239

−0:3247

−0:02300:0230

−0:02390:0240

0:328

0:3592

0:156

0:1697

—

0:5342

0:513

0:5133

503D

LM4(IS2)

0:539−0:5390:5673

−0:5674

0:276−0:2760:2894

−0:2895

−0:02160:0216

−0:02250:0225

0:337

0:3665

0:141

0:1533

—

0:5305

0:446

0:4449

1003D

LM4(IS2)

0:539−0:5390:5655

−0:5655

0:271−0:2710:2841

−0:2841

−0:02140:0214

−0:02240:0224

0:339

0:3665

0:139

0:1505

—

0:5302

0:435

0:4348

1000 LM4(IS2) 0:5646−0:5646

0:2823−0:2823

−0:02230:0223 0:3663 0:1496 0:5301 0:4315

the plate thickness. It should be noticed that a plate consisting of one single layer has beeninvestigated. Good performance of the implemented elements should be con�rmed for localvalue evaluations. The three integration schemes are also compared in Table V: very thick(a=h=2), moderately thick (a=h=4) and thin plates are considered. The plate being made ofone layer, the quoted comparisons are of particular interest, as far as numerical behaviour isconcerned:

• The plate being made of one layer, RMVT and PVD statements must lead to the samenumerical result. This is demonstrated by the fact that the two closed-form solutionsLM4a and LD4a give the same numbers.

• Such a coincidence is preserved in the �nite element solutions.• Furthermore, PVD and RMVT �nite element results are the same for the three di�erentintegration schemes (IN, IS, IS2) that have been implemented.



Ne

a

Uz

2.45

2.5

2.55

2.6

2.65

2.7

2.75

2.8

2.85

2 4 6 8 10 12

Q8Q9

Q4

LD1

Figure 6. �Uz(a=2; a=2; 0) vs Ne. Convergence to the analytical solution. Data: a=h=4, LD1,IS integration, LAM2, sinusoidal load, SS.

Ne

Uza

LM1

2.45

2.5

2.55

2.6

2.65

2.7

2.75

2.8

2.85

2.9

2 3 4 5 6 7 8 9 10

Q8Q4

Q9

Figure 7. �Uz(a=2; a=2; 0) vs Ne. Convergence to the analytical solution. Data: a=h=4,LM1, IS integration, LAM2, sinusoidal load, SS.

Uz

a

2.78

2.8

2.82

2.84

2.86

2.88

2.9

1

N

Q9

3DLDN

42 3

Figure 8. �Uz(a=2; a=2; 0) vs N . Convergence at the analytical solution. Data: a=h=4, LDNcase, IS, LAM2, sinusoidal load, SS.



Ne

Uz

a

0.51

0.511

0.512

0.513

0.514

2 3 4 5 6 7

LD2(IS)LD2(IS2)

LD2

8

Figure 9. �Uz(a=2; a=2; 0) vs Ne. Convergence to the analytical solution. Data: a=h=100,LD2 case, LAM2, sinusoidal load, SS.

Uz

Ne

a

0.51

0.511

0.512

0.513

0.514

2 3 4 5 6 7

LM2(IS2)LM2(IS)

LM2

8

Figure 10. �Uz(a=2; a=2; 0) vs Ne. Convergence to the analytical solution. Data: a=h=100,LM2 case, LAM2, sinusoidal load, SS.

• These three points permit one to conclude that the connections (which were establishedin Appendix C of Part 1) between sti�ness=compliance terms and di�erent integrationschemes should be considered to be consistent for both PVD and RMVT cases.

Further to what has just been written, the authors have made additional analyses, whichare not documented herein, to compare eigenvalues of �nite element matrices related to theproblem discussed in Table IV. A perfect agreement between the classical and mixed layer-wise �nite elements was con�rmed by this eigenvalue analysis. It should once again be notedthat the mathematical structure of the matrices related to the PVD and RMVT applicationsis completely di�erent. Nevertheless, it leads to the same numbers: in these magic resultsprobably lies the truth of variational statements.The convergence rates of transverse displacements by an increase of Ne of LD2 and LD2

cases for IS and IS2 integration schemes are detailed in Figures 9 and 10. Thin plates are



Ne

Uza

2.859

2.86

2.861

2.862

2.863

2.864

2.865

2.866

2 3 4 5 6 7 8 9 10

LD2(IS)LD2(IS2)

LD2

Figure 11. �Uz(a=2; a=2; 0) vs Ne. Convergence to the analytical solution. Data: a=h=4, LD2case, LAM2, sinusoidal load, SS.

Uz

Ne

a

2.87

2.875

2.88

2.885

2.89

2 3 4 5 6 7 8 9 10

LM2(IS)LM2(IS2)

LM2

Figure 12. �Uz(a=2; a=2; 0) vs Ne. Convergence to the analytical solution. Data: a=h=4, LM2case, LAM2, sinusoidal load, SS.

treated in the two latter �gures while corresponding thick plates are addressed in Figures 11and 12. The thickness behaviour of in-plane displacements of thick and thin plates are givenin Figures 13–16 for di�erent integration schemes and LD and LM �nite elements.The classical patch test on transverse displacements vs thickness ratio is presented in

Figure 17. Such a test proves that the three di�erent integration schemes all work well toevaluate transverse displacements of very thick and very thin multilayered plates. On thecontrary, quite a di�erent response has been obtained as far as transverse stress evaluationsare concerned, see Figures 18 and 19. In such a case, the use of the IS2 integration schemebecomes mandatory for thin-plate geometries.The e�ects of the number of elements on transverse displacements and transverse stresses

in thin-plate analyses have been addressed in Figures 20–22. The superiority of the IS2integration scheme is con�rmed. It is further demonstrated that the accuracy of the three



Ux

z/h−1

−0.5

0

0.5

1

−0.4 −0.2 0 0.2 0.4

LM4(IN)LM4(IS)

LM4(IS2)

3D

Figure 13. �Ux (x=0) vs z=h. Convergence to the elasticity solution. Data: a=h=4, LM4case, LAM2, sinusoidal load, SS.

z/h

Ux

10

−10

−5

0

5

−0.4 −0.2 0 0.2 0.4

LM4(IN)LM4(IS)

LM4(IS2)

3D

Figure 14. �Ux (x=0) vs z=h. Convergence to the elasticity solution. Data: a=h=10,LM4 case, LAM2, sinusoidal load, SS.

integration schemes increases by the number of elements that increase, that is, the di�erencesbetween the three di�erent schemes is a numerical problem.A more comprehensive evaluation of displacements and stresses for the di�erent layer-wise

theories is given in Tables V and VI. To complete the picture, transverse displacements oflayer-wise and equivalent single-layer �nite elements, which are based on both PVD andRMVT for thin plates, are compared in Table VII. The following �nal comments can bemade on these latter analyses:

• The introduced integration schemes work in the same manner for both classical and mixed�nite elements, that is, the choice made in Appendix C of Part 1 when the di�erent termswere selected, should be considered reasonable.

• Thick and thin plates can be modelled by appropriate selection of integration schemes.• As far as displacement evaluations are concerned, the three integration schemes lead tothe same results. This conclusion cannot be con�rmed as far as stresses are concerned.In such a case, IN and IS evaluations could be completely wrong.



Ux

z/h

−1

−0.5

0

0.5

1

−0.4 −0.2 0 0.2 0.4

LD4(IN)LD4(IS)

LD4(IS2)

3D

Figure 15. �Ux (x=0) vs z=h. Convergence to the elasticity solution. Data: a=h=4, LD4case, LAM2, sinusoidal load, SS.

z/h

Ux

−10

−5

0

5

10

−0.4 −0.2 0 0.2 0.4

LD4(IN)LD4(IS)

LD4(IS2)

3D

Figure 16. �Ux (x=0) vs z=h. Convergence to the elasticity solution. Data: a=h=10,LD4 case, LAM2, sinusoidal load, SS.

The conclusions reached on the analyses presented here have been used in the results thatare discussed in the next paragraphs. Q9 �nite elements in particular are employed in theimplemented FE model.

3.3. Comparison with other results in the literature

A comprehensive comparison with available computational results has been documentedin Tables VIII–XIII. Some of the most relevant multilayered plate elements that have beenproposed in open literature have been considered in these tables. Di�erent geometries, load-ings and boundary conditions are considered. Problems treated by other authors have beencompared with the implemented �nite elements. In a few cases, closed-form solutions havealso been given and, wherever available, three-dimensional solution has been quoted too. For



LM4(IS)LM4(IN)

0

1

2

3

4

5

0 200 400 600 800 1000a/h

LM4(IS2)3D

a/h

Uz

Figure 17. �Uz(a=2; a=2; 0) vs Ne. Convergence to the elasticity solution. Data: LM4 case,LAM3, bi-sinusoidal load, SS.

σzx

a/h−1

−0.5

0

0.5

1

0 200 400 600 800 1000

LM4(IN)LM4(IS)

LM4(IS2)3D

Figure 18. ��zx(0; a=2; 0) vs a=h. Convergence to the elasticity solution. Data: LM4 case,LAM3, bi-sinusoidal load, SS.

σzy

a/h−1

−0.5

0

0.5

1

0 200 400 600 800 1000

LM4(IN)LM4(IS)

LM4(IS2)3D

Figure 19. ��zy(a=2; 0; 0) vs a=h. Convergence at the elasticity solution. Data: LM4,LAM3, bi-sinusoidal load, SS.



σzy

Ne−3

−2

−1

0

1

2

3

1

LM4(IN)LM4(IS)

LM4(IS2)3D

3 42

Figure 20. ��zy(a=2; 0; 0) vs Ne. Convergence to the elasticity solution. Data: a=h=100, LM4,LAM3, bi-sinusoidal load.

Ne

3D

0.2

0.25

0.3

0.35

0.4

0.45

0.5

1

LM4(IN)LM4(IS)

LM4(IS2)

Uz

2 3 4

Figure 21. �Uz(a=2; a=2; 0) vs Ne. Convergence to the elasticity solution. Data: a=h=100,LM4, LAM3, bi-sinusoidal load, SS.

σzx

−3

−2

−1

0

1

2

3

1 3

LM4(IN)LM4(IS)

LM4(IS2)3D

Ne

42

Figure 22. ��zx(0; a=2; 0) vs Ne. Convergence to the elasticity solution. Data: a=h=100, LM4case, LAM3, bi-sinusoidal load, SS.



Table VII. �Uz(a=2; a=2; 0). Comparison of various theories. Data: IS integration,a=h=100, case of Table VI.

Exact 3D value: 0.435LM4 0:43482 LM3 0:43482 LM2 0:43482 LM1 0:43477LD4 0:43482 LD3 0:43482 LD2 0:43481 LD1 0:43390— — EMZC3 0:43708 EMZC2 0:43700 EMZC1 0:43183EMC4 0:43688 EMC3 0:43690 EMC2 0:43577 EMC1 0:43067— — EDZ34 0:43708 EDZ2 0:43698 EDZ1 0:43180ED4 0:43684 ED3 0:43684 ED2 0:43572 ED1 0:43054

Table VIII. �Uz(a=2; a=2; 0). Comparison to available results.Data: a= b, bi-sinusoidal load, LAM3, SS, mesh 3×3. 3D

solution by Pagano [31].

a=h 4 10 20 100

3D 1:937 0:737 0:513 0:435

Results from literatureR-H 1:8937 0:7147 0:5060 0:4343R-F 1:7100 0:6628 0:4912 0:4337P&K 1:8744 0:7185 — 0:4346D&R 1:9530 0:7377 0:5122 0:4333A&S − 0:6693 — —LH&X 1:7095 0:6627 0:4912 0:4337

Present LW analysisLM3a 1:9385 0:7370 0:5130 0:4346LD3a 1:9371 0:7370 0:5130 0:4346LM3(IS) 1:9381 0:7358 0:5114 0:4328LD3(IS) 1:9381 0:7358 0:5114 0:4328

Present ESL analysisEMZC3(IS) 1:9633 0:7443 0:5167 0:4368EMC4(IS) 1:9506 0:7303 0:5121 0:4366EDZ3(IS) 1:9633 0:7441 0:5166 0:4368ED4(IS) 1:9506 0:7272 0:5112 0:4366

those cases in which a three-dimensional solution is not available, reference could be madeto the present LM4 �nite elements.Thick and thin square plates are considered in Table VIII. Early and recent computa-

tional results are compared to the proposed classical and advanced �nite elements. Classical(R–E) Reissner–Mindlin-type and re�ned (R–E and P&K) �nite elements with only displace-ment variables are considered. D-R consists of a hybrid �nite plate=shell element while A-Sconsists of a very recent mixed re-elaboration of FSDT statements. The alternative computa-tional techniques, called di�erential quadrature approach by Liew et al. [15] are also takeninto account in the comparison. These �ve referenced analyses are compared to the present



Table IX. �Uz(a=2; b=2; 0). Comparison to available results.Data: b=3a, bi-sinusoidal load, LAM2, SS, mesh 3×9. 3D

solution by Pagano [4].

a=h 4 10 20 100

3D 2:820 0:919 0:610 0:508

Results from literatureD&R 2.8370 0.920 0.6086 0.5061D-1 2.7172 0.8810 0.5987 0.50721D-2 2.7756 0.9197 0.6098 0.5077R-H 2.6411 0.862 0.5937 0.5070L&S 2.828 0.921 0.611 —IK&T 2.729 0.918 0.609 —P&T 2.73 0.918 0.610 0.508

Present LW analysisLM3a 2:8216 0:9189 0:6095 0:5077LD3a 2:8210 0:9189 0:6095 0:5077LM3(IS) 2:8167 0:9163 0:6072 0:5054LD3(IS) 2:8167 0:9163 0:6072 0:5054

Present ESL analysisEMZC3(IS) 2:8554 0:9182 0:6077 0:5057EMC4(IS) 2:7648 0:8744 0:5953 0:5052EDZ3(IS) 2:8527 0:9182 0:6077 0:5057ED4(IS) 2:7328 0:8686 0:5938 0:5051

Table X. �Uz(a=2; a=2; 0). Comparison to available results.Data: a= b, LAM2, uniform load, SS, mesh 3×3.

a=h 4 10 100

Results from literatureP&K 2:8765 1:0968 0:6713R-H 2:9091 1:0900 0:6705Mindlin 2:6559 1:0211 0:6701

Present LW analysisLM3(IS) 3:0547 1:1593 0:6743LD3(IS) 3:0548 1:1593 0:6743

Present ESL analysisEMZC3(IS) 3:1702 1:1787 0:6806EMC4(IS) 3:1099 1:1272 0:6799EDZ3(IS) 3:1667 1:1788 0:6806ED4(IS) 3:0797 1:1204 0:6798



Table XI. U ′z (a=2; a=2; 0). Comparison to available results.

Data: b= a, LAM2, central point load, SS, mesh 4×4.a=h 4 10 100

Results from literatureP&K 21:709 5:3434 2:1593Mindlin 15:6905 4:3989 2:1177


Table XII. �Uz(a=2; a=2; 0). Comparison to available results.Data: b= a, LAM2, bi-sinusoidal load, CL, mesh 3×3.

a=h 4 10 100


Present LW analysis

LM3(IS) 1:3612 0:3961 0:1072LD3(IS) 1:3594 0:3958 0:1072


layer-wise and equivalent single-layer �nite elements. Both classical and mixed implementa-tions have been given with correspondence to the highest N -values. The following commentscan be made:

• The di�erences in the several theories vanish in thin-plate cases.• Present LW �nite elements are in excellent agreement with three-dimensional solutions.• Among the considered analyses, the present LW �nite elements lead to the best two-dimensional description.

• The present ESL analyses, based on RMVT, are more e�ective than those related to thereferenced re�ned �nite elements.

Correspondent rectangular-plate cases are addressed in Table IX. Apart from the already con-sidered re�ned models (R-H and D-R), the hybrid multilayered elements by Liou and Sun (L&S),



Table XIII. �Uz(a=2; a=2; 0). Comparison to available results.Data: b= a, LAM2, uniform load, CL, mesh 3×3.

a=h 4 10 100


Present LW analysisLM3(IS) 1:9612 0:5589 0:1442LD3(IS) 1:9577 0:5583 0:1442


Table XIV. U ′z (a=2; a=2; 0). Comparison to available results.

Data: b= a, LAM2, central point load, CL, mesh 4×4.a=h 4 10 100



as well as the re�ned models D-1, D-2, P&T, and IK&T (which are based on Ambartsumian–Whitney–Rath–Das theory given in Part 1), are compared. The quoted results con�rm the con-clusions outlined for Table VIII. The superiority of the present ESL �nite elements EMZC3and EDZ3 (based on RMVT and PVD, respectively) over the other available ESL re�nedmodels, such as D-1, D-2, P&T and IK&T is in particular con�rmed. The four latter refer-enced analyses, by discarding normal stresses e�ects, in fact violate Koiter’s recommendationsdiscussed in Section 2 of Part 1. Such a stress is indeed fully retained by the present EMZC3and EDZ3 analyses. In particular, EMZC3 describes an interlaminar continuous transversestress �eld.Further similar analyses, related to di�erent loadings and boundary conditions, are dealt

with in Tables X–XIV. The previously reached conclusions can be con�rmed. These tablescould also be used to assess future, re�ned models for multilayered plates.



Ux

z/h−1

−0.5

0

0.5

1

−0.4 −0.2 0 0.2 0.4

LM4EMZC3

3D

Figure 23. �Ux (x=0) vs z=h. Comparison to the elasticity solution. Data: a=h=4, INintegration, LAM2, sinusoidal load, SS.

z/h

’σxx

−20

−15

−10

−5

0

5

10

15

20

−0.4 −0.2 0 0.2 0.4

LM4EMZC3

3D

Figure 24. �′xx (x= a=2; ) vs z=h. Convergence to the elasticity solution. Data: a=h=4, INintegration, LAM2, sinusoidal load, SS.

3.4. Further results obtained by comparing the implemented classical andadvanced elements

Quite a comprehensive evaluation of the implemented �nite elements as a tool to trace the re-sponse of multilayered plates is given in Figures 23–33. In-plane displacements,in-plane stresses, transverse stresses (both shear and normal components) vs multilayered platethickness plots are given. Thick (a=h=4) and moderately thick plates are considered. Plottedtransverse stresses are those assumed a priori for mixed cases. In-plane stresses are thoseobtained from the corresponding Hooke’s law. Transverse stresses related to PVD formulated�nite elements are those obtained by integration of three-dimensional inde�nite equilibriumequations.The implemented comparisons permit one to make the following comments:

• It is con�rmed that LM4 analyses lead to three-dimensional descriptions of static responseof multilayered plates.



z/h

σzx’

0

0.5

1

1.5

2

2.5

−0.4 −0.2 0 0.2 0.4

LM4EMZC3

3D

Figure 25. �′zx (x=0; ) vs z=h. Convergence to the elasticity solution. Data: a=h=4, INintegration, LAM2, sinusoidal load, SS.

σzz

z/h−0.2

0

0.2

0.4

0.6

0.8

1

1.2

−0.4 −0.2 0 0.2 0.4

LM4EMZC3

3D

Figure 26. ��zz (x= a=2; ) vs z=h. Convergence to the elasticity solution. Data: a=h=4, INintegration, LAM2, sinusoidal load, SS.

• Among the implemented analysis, layer-wise �nite elements (e.g. LM4) are the onlyones that give a priori transverse stresses (Figure 25) and such a description meets thecorresponding three-dimensional, exact analysis with excellent agreement.

• Among the implemented ESL �nite elements, those formulated on RMVT o�er the bestdescriptions. In any case, very thick plates demand the use of layer-wise �nite elements.

• The largest di�erences among the di�erent �nite elements are mostly located at the layerinterfaces.

• �zz can play a predominant role, as it is extremely in�uenced by a=h. It should benoticed that the considered laminated plates barely being transversely anisotropic (i.e.Ez=ET does not vary in the thickness direction) the zig-zag terms uZz could introducean unmotivated constraint (see Figure 33). The zigzag form has, in fact, barely beenexhibited by the LM4 analysis.



σxx

z/h−0.6

−0.4

−0.2

0

0.2

0.4

0.6

−0.4 −0.2 0 0.2 0.4

EMZC3LM4

LD4

ED4ED1

EDZ3EM4

Figure 27. ��xx (x= a=2; y= a=2) vs z=h. Evaluation of di�erent �nite elements. Data:a=h=10, IS, LAM2, bi-sinusoidal load, SS, mesh 5×5.

z/h

σyy

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

−0.4 −0.2 0 0.2 0.4

LM4EMZC3

LD4EM4

EDZ3ED4ED1

Figure 28. ��yy (x= a=2; y= a=2) vs z=h. Evaluation of di�erent �nite elements. Data: a=h=10, ISintegration,LAM2, bi-sinusoidal load, SS, mesh 5×5.

3.5. Final comments on the analyses

Further to the discussion made in the previous subsections, the following comments andconclusions can be drawn on the conducted numerical investigations.

• Advanced layer-wise LM and equivalent single-layer �nite elements, which are basedon RMVT (LM- and EM-types), completely ful�l interlaminar equilibria and accountfor the zig-zag e�ects. These elements a priori furnish transverse shear normal stresseswithout requiring any post-processing procedures. Worst description has been found inthe ESL cases, due to the assumed, non-physical independence of the zig-zag functionfrom the k-layer.



σxy

z/h−0.03

−0.02

−0.01

0

0.01

0.02

0.03

−0.4 −0.2 0 0.2 0.4

LM4EMZC3

LD4EM4

EDZ3ED4ED1

Figure 29. ��xy (x=0; y=0) vs z=h. Evaluation of di�erent �nite elements. Data: a=h=10, IS integration,LAM2, bi-sinusoidal load, SS, mesh 5×5.

σzx

z/h0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

−0.4 −0.2 0 0.2 0.4

LM4EMZC3

LD4EM4

EDZ3ED4ED1

Figure 30. ��zx (x=0; y= a=2) vs z=h. Evaluation of di�erent �nite elements. Data: a=h=10,IS integration, LAM2, bi-sinusoidal load, SS, mesh 5×5.

• LD- and ED-type �nite elements do not a priori ful�l interlaminar continuous transversestresses while zig-zag e�ects can be included in ED cases by considering the relatedEDZ-type theories.

• LM- and LD-type elements are more computationally expensive than the correspondingEM and ED ones; the number of independent variables in fact depends on the num-ber of constitutive layers Nl. On the other hand, the use of layer-wise description isessential for those cases in which an accurate description of �zz and related e�ects isrequired.

• LWM descriptions are more accurate than the corresponding ESLM ones.• Advanced, M mixed descriptions based on RMVT are more accurate than the corre-sponding classical D formulation, based on PVD.



zyσ

z/h0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

−0.4 −0.2 0 0.2 0.4

LM4EMZC3

LD4EM4

EDZ3ED4ED1

Figure 31. ��zy (x= a=2; y=0) vs z=h. Evaluation of di�erent �nite elements. Data: a=h=10,IS integration, LAM2, bi-sinusoidal load, SS, mesh 5×5.

z/h

Uz

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

−0.4 −0.2 0 0.2 0.4

ED1

LM4EMZC3

LD4

EDZ3ED4

EM4

Figure 32. �Uz (x= a=2; y= a=2) vs z=h. Evaluation of di�erent �nite elements. Data:a=h=10, IS integration, LAM2, bi-sinusoidal load, SS, mesh 5×5.

• EM �nite elements are more accurate than ED ones, in other words RMVT is muchmore e�ective for ESLM formulations.

• M mixed analyses do not require any post-processing procedure to obtain transversestresses.

• The order of the used expansion N plays a very important role, especially as far asunsymmetrical laminates are concerned. One should note that the quadratic expansions aremuch more e�ective for unsymmetrical laminates. With an increase of N the di�erencesbetween LM and LD tends to disappear.



z/h

σzz

a/h

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

−0.4 −0.2 0 0.2 0.4

LM4EMZC3

LD4

Figure 33. 1a=h ��zz (x= a=2; y= a=2) vs z=h. Evaluation of di�erent �nite elements. Data:a=h=10, IS integration, LAM2, bi-sinusoidal load, SS, mesh 5×5.

• The zig-zag function already improves the related results: the EDZ analyses are moreaccurate than the ED ones. Advantages also arise from imposing interlaminar continuity,i.e. the EMC results are more accurate than the ED ones.

4. PROPOSED TEST CASES

The companion paper (Part 1) has proposed advanced and classical �nite elements as toolsto describe the static response of multilayered plates with di�erent levels of accuracy andfrom a two-dimensional point of view. Furthermore, the present paper has shown results thathave been obtained by implementing some of the �nite elements proposed in the companionpaper. Nevertheless, the numerical analysis conducted in the previous sections has documentedthat quite a di�erent level of accuracy could be reached by the several implemented �niteelements. Such a level of accuracy can, in fact, vary considerably from element to element.For instance, a three-dimensional description can be obtained by implementing layer-wise�nite elements (e.g. LM4) while thin-plate results of a Kirchho�-type can be obtained byimplementing classical equivalent single layer elements (e.g. ED1).Many other �nite elements exist between these two extreme �nite elements, related to the

highest and lowest level of accuracy. Two ‘test cases’ are then proposed by the authors in Ta-bles XV(a)–XVI(d) and Figures 34 and 35. A layered, moderately thick plate is considered inTables XV(a) and XV(b) where 20 additional �nite element results have been quoted betweenthe LM4 and ED1 analyses. Very thick (a=h=2), thick (a=h=4), moderately thick (a=h=10)and thin sandwich plates are considered in Tables XVI(a) and XVI(b) where results of six�nite elements have been compared. These six elements correspond to most representativeimplemented �nite elements. In-plane displacements vs thickness of the sandwich plate aregiven in Figures 34 and 35; these two �gures are related to the two thickest plates.Tables XV(a)–XVI(d) also represent a summary of the analysis present in this paper. Three-

dimensional results, as well as a few other �nite element results by other authors, have been



Table XV. Test case I. Results for the 22 implemented Q9 �nite elements. Data: a = b,LAM2, bi-sinusoidal load, SS, mesh 5×5, a=h = 10.

��xx ��yy ��xy ��zx ��zy �Uza=2; a=2;±h=2 a=2; a=2;±h=6 0; 0;±h=2 0; a=2; 0 a=2; 0; 0 a=2; a=2; 0

3D0:590

− 0:5900:285

−0:288−0:02890:0289 0:357 0:1228 0:7530

Results from literature

L&S0:580

− 5800:285

−0:289−0:02830:0284 0:367 0:127 0:7546

Moriya0:5759

− 0:57850:2820

−0:2890−0:028610:02878 0:3993 0:1296 0:7512

R-H0:5684— — — 0:1033 — 0:7125

H&L0:5884

− 0:58790:2834

−0:2873−0:028800:02896 0:3627 0:1284 0:7531

Present LW analysis

LM4(IS2)0:5801

− 0:57840:2796

−0:2831−0:02960:0297 0:3626 0:1249 0:7528

LM3(IS2)0:5801

− 0:57840:2797

−0:2831−0:02960:0298 0:3626 0:1249 0:7528

LM2(IS2)0:5796

− 0:57810:2772

−0:2807−0:02960:0297 0:3518 0:1191 0:7524

LM1(IS2)0:5760

− 0:57480:2525

−0:2562−0:02940:0296 0:3615 0:0937 0:7500

LD4(IS2)0:5801

− 0:57840:2796

−0:2831−0:02960:0297 0:3724 0:1623 0:7528

LD3(IS2)0:5801

− 0:57840:2797

−0:2831−0:02960:0297 0:3724 0:1623 0:7528

LD2(IS2)0:5792

− 0:57770:2791

−0:2826−0:02960:0297 0:3711 0:1362 0:7519

LD1(IS2)0:5608

− 0:55980:2740

−0:2776−0:02880:0289 0:3726 0:1338 0:7371

Present ESL analysis

EMZC3(IS)0:5856

− 0:58260:2829

−0:2823−0:00830:0083 0:3884 0:1270 0:7634

EMZC2(IS)0:5685

− 0:56540:2766

−0:2761−0:00760:0076 0:3955 0:1486 0:7503

EMZC1(IS)0:5674

− 0:56580:2747

−0:2741−0:00770:0077 0:3986 0:1546 0:7442

EMC4(IS)0:5787

− 0:57640:2708

−0:2706−0:00840:0083 0:3118 0:1259 0:7313

EMC3(IS)0:5803

− 0:57730:2710

−0:2705−0:00850:0086 0:2687 0:1450 0:7323

EMC2(IS)0:5128

− 0:50970:2417

−0:2411−0:00550:0055 0:1442 0:1124 0:6441

EMC1(IS)0:5106

− 0:50900:2411

−0:2406−0:00510:0051 0:1979 0:0723 0:6449

EDZ3(IS)0:5856

− 0:58250:2834

−0:2828−0:00830:0083 0:3879 0:1491 0:7634



Table XV Continued.

EDZ2(IS)0:5664

−0:56330:2780

−0:2775−0:00750:0075 0:3838 0:1411 0:7487

EDZ1(IS)0:5642

−0:56250:2762

−0:2757−0:00760:0076 0:3777 0:1408 0:7417

ED4(IS)0:5776

−0:57530:2694

−0:2692−0:00830:0083 0:2948 0:1464 0:7268

ED3(IS)0:5783

−0:57530:2687

−0:2681−0:00850:0085 0:2849 0:1445 0:7246

ED2(IS)0:5131

−0:51010:2405

−0:2399−0:00560:0056 0:1671 0:1137 0:6395

ED1(IS)0:5113

−0:50960:2382

−0:2376−0:00550:0055 0:1538 0:1117 0:6313

Table XVI. Test case II. Results for the most representative Q9 �nite elements. Data:sandwich plate, SS, mesh 5×5.

��xx ��yy ��zx ��zy �Uz(a=2; b=2;±1) (a=2; b=2;±1) (0; b=2; 0) (a=2; 0; 0) (a=2; b=2; 0)

(a) a=h=2 case

3D3:278

−2:6530:452

−0:392 0:185 0:139 26:0

Present layer-wise analysis

LM43:2430

−2:62330:4537

−0:3829 0:1897 0:1444 22:103

LM23:2352

−2:61720:4541

−0:3832 0:1853 0:1404 22:103

LD33:2426

−2:62330:4544

−0:3834 — — 22:103


EMZC33:1594

−2:56120:5041

−0:4338 0:2403 0:1600 23:313

EMZC23:1255

−2:52860:5021

−0:4319 0:2489 0:1648 23:298

ED33:0752

−2:50150:4825

−0:4241 — — 21:960

(b) a=h=4 case

3D1:556

−1:5120:260

−0:253 0:239 0:107 8:0


LM41:5425

−1:49560:2582

−0:2486 0:2459 0:1143 7:5947

LM21:5412

−1:49440:2584

−0:2487 0:2435 0:1118 7:5943

LD31:5426

−1:49570:2583

−0:2487 — — 7:5948



Table XVI Continued.

��xx ��yy ��zx ��zy �Uz(a=2; b=2;±1) (a=2; b=2;±1) (0; b=2; 0) (a=2; 0; 0) (a=2; b=2; 0)


EMZC31:5932

−1:54670:2804

−0:2713 0:2759 0:1192 7:8723

EMZC21:5852

−1:53880:2801

−0:2709 0:2767 0:1186 7:8689

ED31:5645

−1:52280:2656

−0:2592 — — 7:3560

(c) a=h=10 case

3D1:153

−1:1520:110— 0:3 0:053 2:5


LM41:1323

−1:12930:1093

−0:1076 0:3042 0:05354 2:2001

LM21:1323

−1:12930:1093

−0:1077 0:3032 0:05237 2:2001

LD31:1324

−1:12930:1093

−0:1076 — — 2:2001


EMZC31:1488

−1:14670:1137

−1:127 0:3313 0:06388 2:2342

EMZC21:1588

−1:15680:1146

−0:1135 0:3264 0:06765 2:2313

ED31:5645

−1:52280:2656

−0:2592 — — 2:1132

(d) a=h=50 case3D ±1:099 ±0:057 0:323 0:031 1:10

Present layer-wise analysisLM4 ±1:0786 ±0:0559 0:3287 0:0312 0:9343LM2 ±1:0785 ±0:0558 0:3280 0:0305 0:9343LM1 ±1:0791 ±0:0563 0:3281 0:0305 0:9343LD3 ±1:0785 ±0:0558 — — 0:9343

Present ESL analysisEMZC3 ±1:0714 ±0:0572 0:4094 0:1055 0:9354EMZC2 ±1:0843 ±0:0589 0:4672 0:0795 0:9303ED3 ±1:0706 ±0:0569 — — 0:9297

quoted for comparison purposes. Transverse displacement as well as in-plane and out-of-planestress values are compared.The authors propose to refer to the two introduced test cases as desk-beds in order to

establish the level of accuracy of multilayered �nite elements to evaluate stresses and dis-placements of multilayered plates. In other words, the authors propose that any new FE



Z/h

Ux

ED3

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

−0.4 −0.2 0 0.2 0.4

EMZC3

LM2 LM1

LM4 LD3

Figure 34. Test case II. In-plane displacement �Ux vs thickness. Very thick sandwich plate a=h=2.

Z/h

Ux

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

−0.4 −0.2 0 0.2 0.4

LM1LM2 LM4 LD3

EMZC3 ED3

Figure 35. Test case II. In-plane displacement �Ux vs thickness. Thick sandwich plate a=h=4.

contribution directed to a better understanding of multilayered structures and which proposesa ‘re�ned’, ‘advanced’ or ‘improved’ multilayered �nite element should clearly state the levelof accuracy that such a �nite element assumes in the hierarchy that has been established inTables XV(a)–XV(d) and Figures 34 and 35.

5. CONCLUDING REMARKS

This paper has reported results of a numerical investigation directed to implement advanced(based on Reissner’s mixed variational theorem) and classical (based on the principle ofvirtual displacements) multilayered �nite elements that have been proposed in the companionpaper. Linear up to fourth-order displacement and transverse stress �elds have been consideredin the layer=plate thickness. Both layer-wise and equivalent single-layer descriptions havebeen addressed, leading to 22 possible two-dimensional models. Three �nite elements (four,



eight and nine nodes) have been considered. As a result, 22×3 �nite elements have beenimplemented and compared. A condensed indicial notation has been used for implementationpurpose.Cross-ply symmetrical and unsymmetrical multilayered plates, as well as sandwich plates

subject to various loadings and boundary conditions, have been analysed. A numerical assess-ment has been made. Comparisons to available three-dimensional solutions, to correspondinganalytical solutions and other available computational implementations have been made.From a numerical point of view, the following conclusion can be made:

• The implemented Q4, Q8 and Q9 elements have shown convergence rates typical of thatclass of �nite elements.

• Advanced multilayered plate elements based on RMVT have shown the same numericalbehaviour as corresponding classical elements based on PVD, that is, RMVT does notintroduce any further numerical complications than those usually obtained from standardPVD applications.

• In order to contrast shear locking, a choice has been made, with respect to reducedand selective integration schemes, on the sti�ness/compliance contributions of the �niteelement matrices related to RMVT applications. Numerical tests have shown that such achoice was consistent to that made for PVD cases.

As far as the two-dimensional point of view is concerned, the following conclusions can bedrawn:

• The a priori ful�lment of interlaminar continuity for �zz makes mixed models moreattractive than other available models that violate such a continuity.

• Thick-plate analysis has shown that any re�nements of existing theories can be meaning-less unless both interlaminar continuous transverse shear and normal stresses are takeninto account in a re�ned theory.

• The number of independent variables should conveniently be taken as being dependenton Nl in very thick-plate analysis.

• It has been con�rmed that the Reissner mixed variational theorem remains a valuabletool to analyse multilayered plates.

• Layer-wise analyses furnish a quasi-three-dimensional description of displacement andstress �elds even though very thick plates are considered (a=h¿2). Transverse stressevaluations do not require any post-processing (a posteriori) procedure such as the in-tegration of three-dimensional inde�nite equilibrium equations. These stresses are in factcomputed a priori and with excellent accuracy directly by the assumed models.

• The accuracy given by equivalent single layer models is extremely subordinate to theorder of the used expansion in the displacement and stress �elds, to laminate layoutsand to geometrical parameters. The case of cubic displacement �eld (EMZC3) has ledto the best results.

Two ‘Test cases’ have been described; these have been proposed as desk-beds to assess future,‘re�ned’ multilayered plate elements. The presented �nite element evaluations have a strongpractical interest in many problems. In general, accurate stress–strain evaluations are requiredto detect a certain damage in the structure. As a further example, the recent ‘Meyer–Piennig[16] test case’ related sandwich structures have to be mentioned. In this case, LW descriptionis required even though a thin structure is considered [17].



Future works should be directed to a comparison of whole implemented �nite elements tosolve practical problems related to multilayered structures. More complex geometries as wellas multilayered layouts, boundary and loading conditions (including dynamic loadings) shouldbe considered in these analyses. Further attempts should consider alternative techniques, suchas assumed strain �eld concept, in order to contrast numerical, locking mechanisms. Thedi�erent techniques described in Part 1 as far as treatment of stress variables is concernedshould be compared too. Extension to non-linear problems could be a further development ofthe present work.

APPENDIX A: FORTRAN FORM OF FUNDAMENTAL NUCLEUS

A.1. Kk�s matrix for PVD applications

ccNine FORTRAN statements follow.UD(1,1) = ETSUU(k,�,s) * QPP(k,1,1) * FXX (i,j) + ETSUU(k,�,s) * QPP(k,1,3) * FYX (i,j)

+ETSUU(k,�,s) * QPP(k,1,3) * FXY (i,j) + ETSUU(k,�,s) * QPP(k,3,3) * FYY (i,j)+ETZSZUU(k,�,s) * QNN(k,1,1) * F00(i,j)

UD(1,2) = ETSUU(k,�,s) * QPP(k,1,2) * FXY (i,j) + ETSUU(k,�,s) * QPP(k,2,3) * FYY (i,j)+ETSUU(k,�,s) * QPP(k,1,3) * FXX (i,j) + ETSUU(k,�,s) * QPP(k,3,3) * FYX (i,j)+ETZSZUU(k,�,s) * QNN(k,1,2) * F00(i,j)

UD(1,3) = ETSZUU(k,�,s) * QPN(k,1,3) * FX0(i,j) + ETSZUU(k,�,s) * QPN(k,3,3) * FY0(i,j)+ETZSUU(k,�,s) * QNN(k,1,1) * F0X(i,j) + ETZSUU(k,�,s) * QNN(k,1,2) * F0Y(i,j)

UD(2,1) = ETSUU (k,�,s) * QPP(k,1,2) * FYX(i,j) + ETSUU (k,�,s) * QPP(k,1,3) * FXX(i,j)+ETSUU (k,�,s) * QPP(k,2,3) * FYY(i,j) + ETSUU (k,�,s) * QPP(k,3,3) * FXY(i,j)+ETZSZUU (k,�,s) * QNN(k,1,2) * F00(i,j)

UD(2,2) = ETSUU (k,�,s) * QPP(k,2,2) * FYY(i,j) + ETSUU (k,�,s) * QPP(k,2,3) * FXY(i,j)+ETSUU (k,�,s) * QPP(k,2,3) * FYX(i,j) + ETSUU (k,�,s) * QPP(k,3,3) * FXX(i,j)+ETZSZUU (k,�,s) * QNN(k,2,2) * F00(i,j)

UD(2,3) = ETSZUU(k,�,s) * QPN(k,2,3) * FY0(i,j) + ETSZUU(k,�,s) * QPN(k,3,3) * FX0(i,j)+ETZSUU(k,�,s) * QNN(k,1,2) * F0X(i,j) + ETZSUU(k,�,s) * QNN(k,2,2) * F0Y(i,j)

UD(3,1) = ETSZUU(k,�,s) * QNN(k,1,1) * FX0(i,j) + ETSZUU(k,�,s) * QNN(k,1,2) * FY0(i,j)+ETZSUU(k,�,s) * QNP(k,3,1) * F0X(i,j) + ETZSUU(k,�,s) * QNP(k,3,3) * F0Y(i,j)

UD(3,2) = ETSZUU(k,�,s) * QNN(k,1,2) * FX0(i,j) + ETSZUU(k,�,s) * QNN(k,2,2) * FY0(i,j)+ETZSUU(k,�,s) * QNP(k,3,2) * F0Y(i,j) + ETZSUU(k,�,s) * QNP(k,3,3) * F0X(i,j)

UD(3,3) = ETSUU (k,�,s) * QNN(k,1,1) * FXX(i,j) + ETSUU (k,�,s) * QNN(k,1,2) * FYX(i,j)+ETSUU (k,�,s) * QNN(k,1,2) * FXY(i,j) + ETSUU (k,�,s) * QNN(k,2,2) * FYY(i,j)+ETZSZUU(k,�,s) * QNN(k,3,3) * F00(i,j)

cc

The 3×3 elements of the K�sk matrix are denoted by the array UD. ET...-type arrays denotethe thickness integrals. These are a�ected by � and s indexes which come from the product ofused expansion in zF�Fj. QPP, QPN, QNP, QNN are the Hooke’s law lamina arrays. ET...-typeas well as QPP, QPN, QNP, QNN arrays are all a�ected by layer index k. F...-type arraysare the integral on �. These are a�ected by i and j indexes which come from the productof used shape function (e.g. NiNj, NixNj). By putting the above 3×3 elements in appropri-ate loops with indexes k; �; s; i; j, layer and multilayered sti�ness matrices are obtained. The



four fundamental nuclei related to RMVT applications are treated likewise. Kk�su� is written inwhat follows.

A.2. Kk�su� matrix

ccUS(1,1)=ETZSUS(k,�,s) * F00(i,j)US(1,2)= 0.0US(1,3)=ETSUS (k,�,s) * CPN(k,1,3) * FX0(i,j) +ETSUS (k,�,s) * CPN(k,3,3) * FY0(i,j)US(2,1)= 0.0US(2,2)=ETZSUS(k,�,s) * F00(i,j)US(2,3) = ETSUS (k,�,s) * CPN(k,2,3) * FY0 (i,j) + ETSUS (k,�,s) * CPN(k,3,3) * FX0 (i,j)US(3,1)=ETSUS (k,�,s) * FX0(i,j)US(3,2)=ETSUS (k,�,s) * FY0(i,j)US(3,3)=ETZSUS(k,�,s) * f00(i,j)cc

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Classical and advanced multilayered plate elements based upon PVD and RMVT. Part 2: Numerical...

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